Iwahori-Hecke algebras occur in the representation theory of reductive p-adic groups. Such an algebra can be viewed as a deformation of the group algebra of the associated extended affine Weyl group. In this talk a proof is outlined for the assertion that the periodic cyclic homology of an Iwahori-Hecke algebra is isomorphic to the periodic cyclic homology of the group algebra of the associated extended affine Weyl group. In other words, the periodic cyclic homology stays constant during the deformation. The proof is done by introducing a class of algebra morphisms called "weakly spectrum preserving" morphisms.

The overall point of view is that finite type algebras (and in particular Iwahori-Hecke algebras) should be viewed as non-commutative affine algebraic varieties.

The above is joint work with Victor Nistor.

Reference: P. Baum and V. Nistor, "Periodic cyclic homology of Iwahori-Hecke algebras" , C.R.Acad.Sci.Paris, t.332, Serie 1, 1-6, 2001